It’s one of those rare foreign language words for which there is simply no equivalent word in English, but the closest translation would be: conviviality, coziness, intimacy, that feeling of togetherness you have when you spend time with loved ones.
In a Slate article, the Dutch pursuit of gezellig is described this way:
They carve out cozy, delightful moments anywhere they can find them. They bring their families on candlelit, nighttime boat rides through the canals. They chat with their friends at outdoor cafes as the sun sets. They leave work by 6 every evening. And these are not special, once-in-a-blue-moon treats. This is how they live, all the time.
When so many others are making resolutions—pledging to do more, more, more—I resolve to choose, as often as possible, the gezellig over the busy trap.